arxiv: 2605.07187 · v1 · submitted 2026-05-08 · 🌀 gr-qc

Recognition: 3 theorem links

· Lean Theorem

Probing Gravitational Wave Signatures from Periodic Orbits of Regular Black Holes in Asymptotically Safe Gravity

Arun Kumar , Abolhassan Mohammadi , Sushant G. Ghosh

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Pith reviewed 2026-05-11 02:34 UTC · model grok-4.3

classification 🌀 gr-qc

keywords asymptotically safe gravityregular black holesperiodic orbitsgravitational wavesextreme mass-ratio inspiralszoom-whirl orbitseffective potentialquantum corrections

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The pith

Quantum corrections via a scaling parameter in asymptotically safe gravity modify periodic orbits around regular black holes, producing gravitational wave signals with measurable amplitude modulations and phase shifts.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper calculates how a single quantum correction parameter alters the paths of massive particles in bound and periodic orbits near a regular black hole. It classifies these orbits by their zoom, whirl, and vertex counts and by the ratio of orbital frequencies, showing that the correction enhances whirl behavior and shifts the innermost stable orbit. Using the quadrupole approximation, the authors compute the gravitational waveforms emitted during extreme-mass-ratio inspirals and find that the signals carry detectable modulations and phase differences that grow with the correction strength. These waveforms peak in the millihertz band, where planned space-based detectors are sensitive, and the peak amplitude rises steadily with the correction parameter. If correct, precise observations of such inspirals could place limits on quantum-gravity effects in the strong-field regime.

Core claim

In the spacetime of a regular black hole arising in asymptotically safe gravity, the dimensional scaling parameter ξ modifies the effective potential for timelike geodesics, shifting the innermost stable circular orbit and enhancing the whirl component of periodic orbits classified by the topological integers (z, w, v) and the rational frequency ratio q = ω_φ/ω_r − 1. Within the quadrupole approximation, the gravitational waveforms for extreme-mass-ratio inspirals exhibit amplitude modulations and phase shifts that increase with ξ; the corresponding strain spectra peak in the millihertz range accessible to LISA, Taiji, and TianQin, and the peak strain grows monotonically with ξ.

What carries the argument

The zoom-whirl taxonomy of periodic orbits, defined by the integers (z, w, v) for zooms, whirls, and vertices per radial cycle together with the rational frequency ratio q, which organizes the bound geodesics and determines the structure of the emitted gravitational waves.

Load-bearing premise

The metric is taken as the correct singularity-free solution of asymptotically safe gravity whose only adjustable parameter is ξ, and the quadrupole approximation is assumed to capture the dominant gravitational-wave emission from extreme-mass-ratio inspirals.

What would settle it

A set of observed extreme-mass-ratio inspiral waveforms around a supermassive black hole that show no systematic increase in amplitude modulations or peak strain with any measurable parameter analogous to ξ, while remaining within the millihertz sensitivity window, would falsify the predicted monotonic growth and detectability of the quantum corrections.

Figures

Figures reproduced from arXiv: 2605.07187 by Abolhassan Mohammadi, Arun Kumar, Sushant G. Ghosh.

**Figure 2.** Figure 2: FIG. 2. behaviour of the MBO parameters as functions of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Dependence of ISCO parameters on the quantum pa [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Allowed parameter space ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Behaviour of the rational frequency ratio [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Periodic orbits around RBHASG with [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. GW forms (plus polarization [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Effect of the quantum parameter [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Correlation between orbital segments and GW features for the (1 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Frequency-domain amplitude spectra [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12. Characteristic strain [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. Frequency-domain amplitude spectra [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. Characteristic strain [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗

read the original abstract

We investigate bound and periodic timelike geodesics and their associated gravitational-wave (GW) signatures in the spacetime of a regular black hole arising in asymptotically safe gravity (ASG). The geometry incorporates quantum corrections via a running gravitational coupling, encoded in a dimensional scaling parameter $\xi$, that modifies the near-horizon structure while preserving asymptotic flatness. We derive the effective potential for massive test particles and determine the conditions for stable circular and bound motion as functions of $\xi$, including the shift in the innermost stable circular orbit (ISCO). The three topological integers $(z,w,v)$, which represent the number of zooms, whirls, and vertices per radial cycle, are used to categorize the test particles' periodic orbits using Levin's zoom -- whirl taxonomy. Moreover, we employ the rational frequency ratio $q = \frac{\omega_\phi}{\omega_r} - 1$ to find closed orbits, where $\omega_\phi$ and $\omega_r$ stand for the azimuthal and radial frequencies, respectively. We examine how the orbital frequency spectrum is altered, whirl behaviour is enhanced, and deviations from the Schwarzschild limit are produced by the quantum parameter $\xi$. The GW forms for extreme mass-ratio inspirals (EMRIs) are calculated within the quadrupole approximation. We find that as $\xi$ increases, the signals that are released exhibit detectable amplitude modulations and phase shifts. The corresponding typical strain spectra fall within the anticipated sensitivity limits of space-based detectors such as LISA, Taiji, and TianQin, as they peak in the millihertz frequency band. Peak strain increases monotonically with $\xi$, indicating that observational restrictions on quantum-gravity-induced deviations from classical general relativity in the strong-field domain can be obtained from precise measurements of zoom -- whirl dynamics in EMRIs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Applies standard zoom-whirl geodesics and quadrupole waves to an ASG regular black hole, with concrete ξ dependence but fixed-orbit waveforms that skip radiation reaction.

read the letter

This paper works out periodic timelike geodesics in the regular black hole metric from asymptotically safe gravity, classifies them with Levin's (z,w,v) integers and rational frequency ratio q, then generates quadrupole gravitational wave signals from those orbits. The main new element is the explicit dependence of orbital frequencies, ISCO location, whirl strength, and peak strain amplitude on the parameter ξ, which had not been reported for this metric before. They recover the Schwarzschild case at ξ=0 and show that larger ξ enhances whirls and raises the strain monotonically, with spectra peaking in the millihertz band. The effective potential and bound-orbit conditions are derived in a straightforward way that looks reproducible from the equations given. The orbit taxonomy follows established methods without visible mistakes, and the deviations from classical gravity are presented clearly enough to be useful for comparison. The soft spot is the waveform modeling. The signals come from the quadrupole formula applied to fixed, non-evolving periodic geodesics. Real EMRIs radiate and inspiral adiabatically over thousands of cycles, so the observable signal includes frequency drift and cumulative phase set by radiation reaction. Fixed-orbit results miss that evolution, which weakens the claims that amplitude modulations and phase shifts would be detectable or allow tight bounds on ξ with LISA, Taiji, or TianQin. The quadrupole approximation itself may be reasonable for rough estimates but needs justification for the strong-field regime here. This is for specialists in quantum-gravity phenomenology and EMRI gravitational waves who want concrete numbers for one specific modified metric. A reader building catalogs of possible deviations or testing analysis pipelines will find the orbit results worth looking at, even if the detectability section stays preliminary. The geodesic part shows clear thinking and honest use of the literature. I would send it to peer review. The calculation is solid enough to deserve referee time, with the main request being to qualify or extend the waveform claims.

Referee Report

2 major / 2 minor

Summary. The manuscript studies bound and periodic timelike geodesics in the regular black-hole spacetime of asymptotically safe gravity, which incorporates a running gravitational coupling via the dimensional parameter ξ. It derives the effective potential, identifies conditions for stable circular and bound orbits (including the ISCO shift), classifies periodic orbits via the zoom-whirl integers (z, w, v) and the rational frequency ratio q = ω_φ/ω_r − 1, and computes the associated gravitational waveforms in the quadrupole approximation for extreme-mass-ratio inspirals. The central claim is that increasing ξ produces detectable amplitude modulations and phase shifts whose peak strains lie in the millihertz band accessible to LISA, Taiji, and TianQin, with peak strain rising monotonically with ξ.

Significance. If the results on orbit classification and the monotonic strain dependence hold after addressing the modeling limitations, the work would supply a concrete, falsifiable signature of quantum-gravity corrections in the strong-field regime that could be tested with future space-based detectors. The structured use of Levin’s zoom-whirl taxonomy and the explicit mapping to frequency ratio q provide a useful framework for exploring deviations from the Schwarzschild case.

major comments (2)

[GW waveforms section] The gravitational-wave forms for EMRIs are obtained by applying the quadrupole formula directly to fixed periodic geodesics classified by (z, w, v) and q (see the section on GW forms and the associated figures). This fixed-orbit treatment omits the radiation-reaction force and the adiabatic inspiral that cause the orbit to traverse a sequence of zoom-whirl configurations while accumulating phase over ∼10^4–10^5 cycles. Consequently, the reported amplitude modulations, phase shifts, and monotonic rise in peak strain do not correspond to the observable EMRI signal; the cumulative phase evolution and frequency drift that determine detectability and distinguishability from GR are absent.
[results on strain spectra] The abstract and results assert that peak strain increases monotonically with ξ and that the spectra fall within LISA/Taiji/TianQin sensitivity, yet no explicit derivation, numerical tabulation, or error estimate supporting the monotonicity is referenced. Because this monotonic dependence is presented as the key observable signature for constraining ξ, its evidential basis must be shown explicitly (e.g., via an equation relating strain amplitude to ξ or a table of computed values).

minor comments (2)

[effective potential derivation] The abstract states that the effective potential, orbit conditions, and frequency ratio q are derived, but the manuscript should ensure every intermediate step (including the explicit form of the effective potential as a function of ξ) appears with numbered equations for reproducibility.
[periodic orbits section] Notation for the three integers (z, w, v) and the frequency ratio q should be introduced with a brief reminder of Levin’s taxonomy in the main text, even if a reference is given, to aid readers unfamiliar with the zoom-whirl classification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight important limitations in our modeling of EMRI waveforms and the presentation of numerical results. We address each point below and will revise the manuscript accordingly to improve clarity and rigor while preserving the core findings on periodic orbits and their quadrupole signatures.

read point-by-point responses

Referee: The gravitational-wave forms for EMRIs are obtained by applying the quadrupole formula directly to fixed periodic geodesics classified by (z, w, v) and q (see the section on GW forms and the associated figures). This fixed-orbit treatment omits the radiation-reaction force and the adiabatic inspiral that cause the orbit to traverse a sequence of zoom-whirl configurations while accumulating phase over ∼10^4–10^5 cycles. Consequently, the reported amplitude modulations, phase shifts, and monotonic rise in peak strain do not correspond to the observable EMRI signal; the cumulative phase evolution and frequency drift that determine detectability and distinguishability from GR are absent.

Authors: We agree that the present calculation applies the quadrupole formula to fixed periodic geodesics and therefore does not incorporate radiation-reaction forces or the adiabatic inspiral through a sequence of orbits. This fixed-orbit approach is adopted to isolate the direct imprint of the quantum parameter ξ on individual zoom-whirl waveforms, including the enhanced whirl behavior and resulting amplitude modulations. While these features would appear as building blocks within a full radiation-reaction-driven inspiral, the cumulative phase accumulation and frequency drift are indeed absent from our current results. We will revise the GW waveforms section and the abstract to explicitly state this approximation and its implications, noting that the reported modulations indicate potential strong-field deviations that merit further study with radiation-reaction included. A complete adiabatic inspiral calculation lies beyond the scope of this work. revision: partial
Referee: The abstract and results assert that peak strain increases monotonically with ξ and that the spectra fall within LISA/Taiji/TianQin sensitivity, yet no explicit derivation, numerical tabulation, or error estimate supporting the monotonicity is referenced. Because this monotonic dependence is presented as the key observable signature for constraining ξ, its evidential basis must be shown explicitly (e.g., via an equation relating strain amplitude to ξ or a table of computed values).

Authors: The monotonic rise in peak strain with increasing ξ was obtained from direct numerical evaluation of the quadrupole waveforms across a range of ξ values while holding orbital parameters fixed. We will add a new table to the results section that tabulates the computed peak strain amplitudes for representative values of ξ (including the Schwarzschild limit ξ = 0), together with a short explanatory paragraph describing how the modified effective potential and orbital frequencies lead to this dependence. This addition will supply the explicit numerical evidence requested. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper takes the ASG metric (with free parameter ξ) as an external input, derives the effective potential and timelike geodesic equations in the standard way, computes orbital frequencies ω_φ and ω_r to obtain the rational ratio q and the (z,w,v) classification, and then applies the quadrupole formula to produce strains. None of these steps reduces by the paper's own algebra to a self-definition, a fitted quantity relabeled as a prediction, or a load-bearing self-citation whose content is itself unverified. The reported trends (amplitude modulations, phase shifts, monotonic rise in peak strain with ξ) are direct numerical consequences of the modified metric inside the geodesic and quadrupole expressions; they are not forced by construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 1 invented entities

The central claim rests on the validity of the ASG regular black-hole metric, the assumption that test particles follow timelike geodesics, and the adequacy of the quadrupole formula for EMRIs; ξ is treated as an external input rather than a fitted constant.

free parameters (1)

ξ
Dimensional scaling parameter that encodes the running of the gravitational coupling; it is varied parametrically rather than fitted to data.

axioms (3)

domain assumption The given line element is a regular, asymptotically flat solution of asymptotically safe gravity
Invoked in the opening paragraph to justify the background spacetime.
standard math Test particles follow timelike geodesics derived from the metric
Standard assumption in general-relativistic orbital dynamics.
domain assumption Gravitational-wave emission is adequately described by the quadrupole formula
Used explicitly for the EMRI waveforms.

invented entities (1)

Regular black hole with running gravitational coupling no independent evidence
purpose: To replace the classical singularity with a quantum-corrected core while preserving asymptotic flatness
The metric is taken from the ASG literature; no independent falsifiable prediction for its existence is supplied in the paper.

pith-pipeline@v0.9.0 · 5641 in / 1827 out tokens · 65966 ms · 2026-05-11T02:34:54.450034+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction contradicts

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contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

the exterior metric is static and spherically symmetric, with the lapse function modified by a logarithmic term involving the scale parameter ξ... 0 < ξ < 0.4565 M²
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel contradicts

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contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

We employ the rational frequency ratio q = ωϕ/ωr − 1 to find closed orbits... GW forms for extreme mass-ratio inspirals (EMRIs) are calculated within the quadrupole approximation
IndisputableMonolith/Foundation/Constants.lean phi_golden_ratio contradicts

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contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

Peak strain increases monotonically with ξ, indicating that observational restrictions on quantum-gravity-induced deviations... can be obtained

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